Remark 14 We can also write \(\frac{1}{2} D_n^1\)  and \(D_n^3\)  as Fredholm determinants with kernels different from \(K_{sp}, K_o\)  via the particle–hole involution of e.g. [1]} since upper bounds on the length of the partition \(\ell (\lambda )\)  translate into lower bounds on the leftmost hole (absence of a particle) in the process \(\lbrace  \lambda _i - i + 1/2 \rbrace \)  and the latter are Fredholm determinants themselves.

